Singularities of ball quotients

TL;DR

This paper investigates the nature of singularities in ball quotients and their toroidal compactifications, establishing conditions under which these spaces have canonical singularities based on dimension and lattice properties.

Contribution

It provides new criteria for when ball quotients and their compactifications possess canonical singularities, extending previous understanding in complex hyperbolic geometry.

Findings

Ball quotients have canonical singularities under certain dimension and lattice restrictions.

Extension of singularity results to toroidal compactifications.

Conditions identified for the type of singularities in complex hyperbolic quotients.

Abstract

We prove a result on the singularities of ball quotients $\Gamma\backslash\CC H^n$. More precisely, we show that a ball quotient has canonical singularities under certain restrictions on the dimension $n$ and the underlying lattice. We also extend this result to the toroidal compactification $(\Gamma\backslash\CC H^n)^*$.